# Mathematica Moravica, Vol. 13-1 (2009)

Some Results About Best and 2-Best Approximation on 2-Structures
Mathematica Moravica, Vol. 13-1 (2009), 1–11.

Abstract. The problem of best approximation has been studied by many mathematicians. Most of these works have dealt with the existence, uniqueness and characterization of best approximations in spaces of continuous functions with values in Banach spaces. But little or no work on approximation has been done on 2-structures such 2-normed spaces, generalized 2-normed spaces and 2-Banach spaces. It is the aim of this paper to investigate the above two concepts in the sense of latter spaces. It is also to investigate the uniqueness and to give attention of this subject from this view.
Keywords. 2-normed space, best approximation, 2-best approximation, 2-Banach space.

On a Mixed Problem for Navier-Stokes System in the Unit Cube
Mathematica Moravica, Vol. 13-1 (2009), 13–24.

Abstract. Mixed problems on the unit balls have special complexity. By using a new method we shall give some sufficient conditions for existence of solutions of the Fefferman's problem B ([4]).
Keywords. Navier-Stokes equations, potential theory, boundary conditions, unit ball.

Power Semigroups that are 0-Archimedean
Mathematica Moravica, Vol. 13-1 (2009), 25–28.

Abstract. In this paper we give a structural characterization for semigroups whose power semigroups are 0-Archimedean. We prove that these semigroups are exactly the nilpotent semigroups.
Keywords. Power semigroup, 0-Archimedean semigroup, homogroup, nilpotent semigroup.

On $e^{\ast}$-Open Sets and $(\mathcal{D},\mathcal{S})^{\ast}$-Sets
Mathematica Moravica, Vol. 13-1 (2009), 29–36.

Abstract. The aim of this paper is to introduce some new classes of sets and some new classes of continuity namely $e^{\ast}$-open sets, $(\mathcal{D},\mathcal{S})$-sets, $(\mathcal{DS},\mathcal{E})$-sets, $(\mathcal{D},\mathcal{S})^{\ast}$-sets, $(\mathcal{DS},\mathcal{E})^{\ast}$-sets, $e^{\ast}$-continuity, $(\mathcal{D},\mathcal{S})^{\ast}$-continuity and $(\mathcal{DS},\mathcal{E})^{\ast}$$-continuity. Properties of these new concepts are investigated. Moreover, some new decompositions of continuity are provided. Keywords.$e^{\ast}$-open set,$e^{\ast}$-continuity, decomposition of continuity,$(\mathcal{D},\mathcal{S})$-set,$(\mathcal{DS},\mathcal{E})$-set,$(\mathcal{D},\mathcal{S})^{\ast}$-set,$(\mathcal{DS},\mathcal{E})^{\ast}$-set,$(\mathcal{D},\mathcal{S})^{\ast}$-continuity,$(\mathcal{DS},\mathcal{E})^{\ast}$-continuity. Some New 2-Designs from a Wreath Product on 18 Points Mathematica Moravica, Vol. 13-1 (2009), 37–41. Download PDF file: (341kB) | Abstract and keywords Abstract. The total of 22 2-designs on 18 points have been found. All these designs have the same group as an automorphism group. This group can be represented as the wreath product of$G$and$H$, where$G$denotes the subgroup of order 3 of PSL(2,2) and$H$denotes the subgroup of order 12 of PSL(2,5). The$2$-$(18,7,42\cdot s)$designs for$s\in\{15, 19, 25, 27, 30, 37, 38, 50\}$and the$2$-$(18,8,28\cdot s)$designs for$s\in\{27, 44, 46, 48, 50, 53, 54, 57, 59, 61, 73, 77, 80, 81\}$have been detected. Up to our knowledge, 16 of these 22 found designs are new. Keywords. Designs, wreath product, automorphism group. A Remark on the Moduli of the Roots of Algebraic Equations Mathematica Moravica, Vol. 13-1 (2009), 43–47. Download PDF file: (279kB) | Abstract and keywords Abstract. In this paper we obtain several upper bounds of the moduli of the roots of the algebraic equations. Keywords. Roots of algebraic equations, upper bounds for roots moduli. Transversal Ordered Interval and Edges Spaces, Fixed Points and Applications Mathematica Moravica, Vol. 13-1 (2009), 49–75. Download PDF file: (566kB) | Abstract and keywords Abstract. In this paper we formulate a new structure of spaces which we call it transversal (upper, middle or lower) ordered interval spaces. Also, we formulate a new structure of spaces which we call it transversal (upper, middle or lower) ordered edges spaces. We introduce this concepts as a natural extension of transversal probabilistic, Fréchet's, Kurepa's and Menger's spaces. This are concepts of transversal spaces with nonnumerical transverses. Transversal ordered interval and edges spaces are new concepts of spaces in the fixed point theory and further a new way in nonlinear functional analysis. In this sense , we introduce notions of the ordered interval contractions on upper and lower transversal ordered interval spaces and prove some fixed point statements as and further applications. This concept have very important applications in numerical analysis and quantum particle physics by L. Collatz [Funktionalanalysis und Num. Math. Springer-Verlag, 1964]. Keywords. General Ecart; distance, Fréchet's spaces, Kurepa's spaces, Menger's spaces, transversal spaces, nonnumerical transverses, transversal probabilistic spaces, transversal ordered interval spaces, transverse, bisection functions, fixed points, ordered interval cont. New Equivalents of the Axiom of Choice and Consequences Mathematica Moravica, Vol. 13-1 (2009), 77–94. Download PDF file: (220kB) | Abstract and keywords Abstract. This paper continues the study of the Axiom of Choice by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Tasković [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Set Integral Equations in Metric Spaces Mathematica Moravica, Vol. 13-1 (2009), 95–102. Download PDF file: (349kB) | Abstract and keywords Abstract. Let$P_{cp,cv}(\mathbb{R}^{n})$be the family of all nonempty compact, convex subsets of$\mathbb{R}^{n}$. We consider the following set integral equations: $X(t) = \int_{a}^{b} K(t,s,X(s))\mathrm{d}s + X_{0},$ $X(t) = \int_{a}^{t} K(t,s,X(s))\mathrm{d}s + X_{0},$ where$K: [a, b] \times [a, b] \times P_{cp,cv}(\mathbb{R}^{n}) \to P_{cp,cv}(\mathbb{R}^{n})$and$X_{0}\in P_{cp,cv}(\mathbb{R}^{n})\$.